On squares in Lucas sequences

Let P and Q be non-zero integers. The Lucas sequence U_n(P,Q) is defined by U_0=0, U_1=1, U_n= P*U_{n-1}-Q*U_{n-2} for n >1. The question of when U_n(P,Q) can be a perfect square has generated interest in the literature. We show that for n=2,...,7, U_n is a square for infinitely many pairs (P,Q) with gcd(P,Q)=1; further, for n=8,...,12, the only non-degenerate sequences where gcd(P,Q)=1 and U_n(P,Q)=square, are given by U_8(1,-4)=21^2, U_8(4,-17)=620^2, and U_12(1,-1)=12^2.Comment: 11 pages. To appear in Journal of Number Theor

Coding of the Reach Vector in Parietal Area 5d

Competing models of sensorimotor computation predict different topological constraints in the brain. Some models propose population coding of particular reference frames in anatomically distinct nodes, whereas others require no such dedicated subpopulations and instead predict that regions will simultaneously code in multiple, intermediate, reference frames. Current empirical evidence is conflicting, partly due to difficulties involved in identifying underlying reference frames. Here, we independently varied the locations of hand, gaze, and target over many positions while recording from the dorsal aspect of parietal area 5. We find that the target is represented in a predominantly hand-centered reference frame here, contrasting with the relative code seen in dorsal premotor cortex and the mostly gaze-centered reference frame in the parietal reach region. This supports the hypothesis that different nodes of the sensorimotor circuit contain distinct and systematic representations, and this constrains the types of computational model that are neurobiologically relevant

Alternating quaternary algebra structures on irreducible representations of sl(2,C)

We determine the multiplicity of the irreducible representation V(n) of the simple Lie algebra sl(2,C) as a direct summand of its fourth exterior power $\Lambda^4 V(n)$. The multiplicity is 1 (resp. 2) if and only if n = 4, 6 (resp. n = 8, 10). For these n we determine the multilinear polynomial identities of degree $\le 7$ satisfied by the sl(2,C)-invariant alternating quaternary algebra structures obtained from the projections $\Lambda^4 V(n) \to V(n)$. We represent the polynomial identities as the nullspace of a large integer matrix and use computational linear algebra to find the canonical basis of the nullspace.Comment: 26 pages, 13 table

Simulating Hamiltonian dynamics using many-qudit Hamiltonians and local unitary control

When can a quantum system of finite dimension be used to simulate another quantum system of finite dimension? What restricts the capacity of one system to simulate another? In this paper we complete the program of studying what simulations can be done with entangling many-qudit Hamiltonians and local unitary control. By entangling we mean that every qudit is coupled to every other qudit, at least indirectly. We demonstrate that the only class of finite-dimensional entangling Hamiltonians that aren't universal for simulation is the class of entangling Hamiltonians on qubits whose Pauli operator expansion contains only terms coupling an odd number of systems, as identified by Bremner et. al. [Phys. Rev. A, 69, 012313 (2004)]. We show that in all other cases entangling many-qudit Hamiltonians are universal for simulation

Quantum Sampling Problems, BosonSampling and Quantum Supremacy

There is a large body of evidence for the potential of greater computational power using information carriers that are quantum mechanical over those governed by the laws of classical mechanics. But the question of the exact nature of the power contributed by quantum mechanics remains only partially answered. Furthermore, there exists doubt over the practicality of achieving a large enough quantum computation that definitively demonstrates quantum supremacy. Recently the study of computational problems that produce samples from probability distributions has added to both our understanding of the power of quantum algorithms and lowered the requirements for demonstration of fast quantum algorithms. The proposed quantum sampling problems do not require a quantum computer capable of universal operations and also permit physically realistic errors in their operation. This is an encouraging step towards an experimental demonstration of quantum algorithmic supremacy. In this paper, we will review sampling problems and the arguments that have been used to deduce when sampling problems are hard for classical computers to simulate. Two classes of quantum sampling problems that demonstrate the supremacy of quantum algorithms are BosonSampling and IQP Sampling. We will present the details of these classes and recent experimental progress towards demonstrating quantum supremacy in BosonSampling.Comment: Survey paper first submitted for publication in October 2016. 10 pages, 4 figures, 1 tabl

Simultaneous Arithmetic Progressions on Algebraic Curves

A simultaneous arithmetic progression (s.a.p.) of length k consists of k points (x_i, y_\sigma(i)), where x_i and y_i are arithmetic progressions and \sigma is a permutation. Garcia-Selfa and Tornero asked whether there is a bound on the length of an s.a.p. on an elliptic curve in Weierstrass form over Q. We show that 4319 is such a bound for curves over R. This is done by considering translates of the curve in a grid as a graph. A simple upper bound is found for the number of crossings and the 'crossing inequality' gives a lower bound. Together these bound the length of an s.a.p. on the curve. We then use a similar method to extend the result to arbitrary real algebraic curves. Instead of considering s.a.p.'s we consider k^2/3 points in a grid. The number of crossings is bounded by Bezout's Theorem. We then give another proof using a result of Jarnik bounding the number of grid points on a convex curve. This result applies as any real algebraic curve can be broken up into convex and concave parts, the number of which depend on the degree. Lastly, these results are extended to complex algebraic curves.Comment: 11 pages, 6 figures, order of email addresses corrected 12 pages, closing remarks, a reference and an acknowledgment adde